You are not wrong that influence can cause those, as any disease that causes a fever can cause multi organ failure and death. The likelihood of the common cold killing a person is however fairly low if the person is young and in otherwise good health.

The criteria I would use is if the medical community is recommending testing for damage after illness. I an pretty use the recommendation is not to test the lungs, kidneys, heart, and brain after an influenza infection for people with no other symptoms. I could be wrong, so feel free to correct me here.

I do know however that dive doctors recommend divers to test for lung damage after a confirmed covid infection, even with no symptoms after infection. There is no such recommendation for influenza, so I would hazard a guess that the risk for otherwise healthy divers after an influenza infection is less than of covid. I have yet see a recommendation for omicron, but for now I would be cautious until the data is in.